| L(s) = 1 | + 1.35·2-s + 2.29·3-s − 0.162·4-s + 4.25·5-s + 3.11·6-s − 1.26·7-s − 2.93·8-s + 2.28·9-s + 5.76·10-s − 1.84·11-s − 0.374·12-s + 3.99·13-s − 1.71·14-s + 9.78·15-s − 3.64·16-s − 17-s + 3.10·18-s − 1.83·19-s − 0.692·20-s − 2.90·21-s − 2.50·22-s − 5.40·23-s − 6.74·24-s + 13.1·25-s + 5.41·26-s − 1.63·27-s + 0.205·28-s + ⋯ |
| L(s) = 1 | + 0.958·2-s + 1.32·3-s − 0.0813·4-s + 1.90·5-s + 1.27·6-s − 0.478·7-s − 1.03·8-s + 0.762·9-s + 1.82·10-s − 0.556·11-s − 0.107·12-s + 1.10·13-s − 0.458·14-s + 2.52·15-s − 0.912·16-s − 0.242·17-s + 0.730·18-s − 0.420·19-s − 0.154·20-s − 0.634·21-s − 0.533·22-s − 1.12·23-s − 1.37·24-s + 2.62·25-s + 1.06·26-s − 0.315·27-s + 0.0389·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.964771127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.964771127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.35T + 2T^{2} \) |
| 3 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 - 4.25T + 5T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 + 5.40T + 23T^{2} \) |
| 29 | \( 1 - 0.208T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 - 7.75T + 37T^{2} \) |
| 41 | \( 1 - 0.528T + 41T^{2} \) |
| 47 | \( 1 + 7.43T + 47T^{2} \) |
| 53 | \( 1 + 9.53T + 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 + 0.882T + 67T^{2} \) |
| 71 | \( 1 - 7.55T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 2.27T + 89T^{2} \) |
| 97 | \( 1 - 0.780T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.06348912723479873611958828071, −9.455928115028257122180673100619, −8.871989220249293064314774693561, −8.015272513449785425020500436275, −6.38450043801248723655932806180, −6.00756202527985186040533488441, −4.93863908466577445469197093871, −3.70107426010151963430647567609, −2.81792584052804078218803378094, −1.93868744237952237366246173391,
1.93868744237952237366246173391, 2.81792584052804078218803378094, 3.70107426010151963430647567609, 4.93863908466577445469197093871, 6.00756202527985186040533488441, 6.38450043801248723655932806180, 8.015272513449785425020500436275, 8.871989220249293064314774693561, 9.455928115028257122180673100619, 10.06348912723479873611958828071
